2021/03/08 by Borovoi, Mikhail, Gornitskii, Andrei A., Rosengarten, Zev
#20G15 #20G20 #Algebraic Geometry (math.AG) #FOS: Mathematics #Group Theory (math.GR) #Primary: 11E72. Secondary: 20G07 #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2103.04654
By a quasi-connected reductive group (a term of Labesse) over an arbitrary field we mean an almost direct product of a connected semisimple group and a quasi-torus (a smooth group of multiplicative type). We show that a linear algebraic group is quasi-connected reductive if and only if it is isomorphic to a smooth normal subgroup of a connected reductive group. We compute the first Galois cohomology set H1(R,G) of a quasi-connected reductive group G over the field R of real numbers in terms of a certain action of a subgroup of the Weyl group on the Galois cohomology of a fundamental quasi-torus of G.